Two DGAs are said to be topologically equivalent when the corresponding Eilenberg–Mac Lane ring spectra are weakly equivalent as ring spectra. Quasi-isomorphic DGAs are topologically equivalent, but the converse is not necessarily true. As a counterexample, Dugger and Shipley showed that there are DGAs that are nontrivially topologically equivalent, ie topologically equivalent but not quasi-isomorphic.
In this work, we define E-infinity topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially E-infinity topologically equivalent E-infinity DGAs. Also, we show using these obstruction
theories that for coconnective E-infinity Fp–DGAs, E-infinity topological equivalences and quasi-isomorphisms agree. For E-infinity Fp–DGAs with trivial first homology, we show that an E-infinity topological equivalence induces an isomorphism in homology that preserves the Dyer–Lashof operations and therefore induces an H-infinity Fp–equivalence.
Advisors/Committee Members: Shipley, Brooke (advisor), Bousfield, Aldridge K (committee member), Antieau, Benjamin (committee member), Gillet, Henri (committee member), Mathew, Akhil (committee member), Shipley, Brooke (chair).